If the device accepts 12 V DC and outputs 220 V AC at 3000 watts, it is called an inverter, NOT a transformer. And 3000 watts delivered by 12 V requires 3000 / 12 = 250 A of current from the 12 V battery at 100% inverter efficiency, which is impossible to achieve.
The
@AnalogKid was being optimistic quoting 90% efficiency. That doesn't include the losses (inefficiencies) of the motor, the drive train, and where the rubber meets the road at 20+ mph. You will be lucky to reach 50% efficiency from battery to wheels-on-the-road. And how will you control the speed of your AC motor? Variable frequency drive? What kind of AC motor are you planning to use? How will you control its speed and torque?
Methinks you don't know WTF you are doing. A 3 kW inverter provides about 4 HP, adequate for a "go kart" type vehicle, but it will very quickly drain any 12 V battery that is sufficiently small to carry on board. Show us your engineering design calculations, specifications and model number for the motor you intend to use, gear-ratio from motor output shaft to driven-wheel axle, estimated weight of vehicle and weight of driver.
Please, no rectal extractions or hand waving for design specifications! Building a vehicle that
safely carries a child (or anyone) close to the ground at 20 mph is not something you do on the spur of the moment, nor "on the fly" with a "let's try this" approach. Steering and braking alone are major design considerations, before even attempting to add a motor with acceleration and speed control. What are your qualifications to proceed with this project?
Inverter technology is popular for use in emergency power supplies, but getting a good sinusoidal wave form out of them is problematical. Many so-called sinusoidal inverters do no such thing, instead providing a triangular wave output, full of harmonics, that does not play well with AC motors. Check inverter specs and output wave forms carefully to make sure the inverter will work with your particular AC motor.
You might want to consider building your son's kart around a commercial, portable, gasoline engine-motor generator set that does produce a reasonable sinusoidal output at constant frequency. At the very least, you could use the MG set to re-charge the battery on your son's go-kart if you are determined to build an all-electric vehicle.